| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2011 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Resultant of three coplanar forces |
| Difficulty | Moderate -0.3 This is a standard M1 mechanics question requiring resolution of forces into components and finding the resultant. While it involves multiple forces and angles, the method is routine: resolve each force into horizontal/vertical (or along AB/perpendicular) components, sum them, then use Pythagoras and trigonometry. The angles are straightforward (30°, 60°, 90°), making calculations cleaner than average. Slightly easier than a typical A-level question due to its procedural nature and nice angles. |
| Spec | 3.03e Resolve forces: two dimensions3.03p Resultant forces: using vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (a) \([2 \times 12\cos 40° - 15\cos 50°]\) | M1 | For resolving in direction \(\overrightarrow{AB}\) |
| Component is 8.74 N | A1 | |
| (b) Component is 11.5 N | B1 | 3 |
| (ii) Magnitude is 14.4 N or direction is \(52.7°\) (or \(0.920°\)) anticlockwise from \(\mathbf{i}\) dir'n | M1 | For using \(R^2 = X^2 + Y^2\) or \(\tan\theta = Y/X\) |
| A1 | ||
| Direction is \(52.7°\) (or \(0.920°\)) anticlockwise from \(\mathbf{i}\) dir'n or magnitude is 14.4 N | B1 | 3 |
| **(i) (a)** $[2 \times 12\cos 40° - 15\cos 50°]$ | M1 | For resolving in direction $\overrightarrow{AB}$ |
| Component is 8.74 N | A1 | |
| **(b)** Component is 11.5 N | B1 | 3 |
| **(ii)** Magnitude is 14.4 N or direction is $52.7°$ (or $0.920°$) anticlockwise from $\mathbf{i}$ dir'n | M1 | For using $R^2 = X^2 + Y^2$ or $\tan\theta = Y/X$ |
| | A1 | |
| Direction is $52.7°$ (or $0.920°$) anticlockwise from $\mathbf{i}$ dir'n or magnitude is 14.4 N | B1 | 3 |
---3\\
\includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_476_714_744_719}
Three coplanar forces of magnitudes $15 \mathrm {~N} , 12 \mathrm {~N}$ and 12 N act at a point $A$ in directions as shown in the diagram.\\
(i) Find the component of the resultant of the three forces
\begin{enumerate}[label=(\alph*)]
\item in the direction of $A B$,
\item perpendicular to $A B$.\\
(ii) Hence find the magnitude and direction of the resultant of the three forces.
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 2011 Q3 [6]}}